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Topic: Lyapunov function

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	Lyapunov stability - Wikipedia, the free encyclopedia
		Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov.
		However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not applicable.
		Lyapunov's realisation was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function can be found to satisfy the above constraints.
	en.wikipedia.org /wiki/Lyapunov_stability (237 words)

	Lyapunov (Site not responding. Last check: 2007-10-20)
		Lyapunov was not the only one being coached by Sechenov who was teaching his own daughter Natalia Rafailovna Sechenov at the same time.
		Lyapunov established that with variation in the angular velocity of revolution Maclaurin ellipsoids pass into Jacobi ellipsoids.
		Topics considered in [8] include: stability, particularly the stability of critical points; the construction and the application of the Lyapunov function; stability of functional- differential equations; the second Lyapunov method; and the method of the Lyapunov vector function in stability theory and nonlinear analysis.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Lyapunov.html (1165 words)

	Nonlinear, Time-Varying DWNs
		Lyapunov stability analysis involves finding a positive-definite ``Lyapunov function'' (which is like a norm defined on the filter state), which decreases each time step (given zero inputs) in the presence of linear or nonlinear computations (such as numerical round-off).
		Whenever each state variable (delay element) in a filter computation has a physical interpretation as an independent wave variable, a Lyapunov function can be defined simply as the sum of wave energies associated with the state variables.
		In practice, Lyapunov stability in a DWN is normally guaranteed by ensuring that wave-variables are never amplified by nonlinear operations or time-varying gains.
	ccrma-www.stanford.edu /~jos/wgj/Nonlinear_Time_Varying_DWNs.html (859 words)

	The Exponential Hash Function
		The problem with the first approach is that the hash functions used in double hashing must be quickly evaluated, yet must also preserve uniform distribution of the hashed data in the table space.
		The function cannot be a linear function of i, or it would suffer from the same limitations as double hashing.
		This function is similar to the RSA and ElGamal cryptosystems [Stinson], in that a finite field exponent is used to create a nonlinear permutation of values.
	www.jea.acm.org /ARTICLES/Vol2Nbr3/node10.html (433 words)

	[No title]
		A plot of the regions where the Lyapunov function exists in parameter space next to a plot over a and b of the largest Lyapunov exponent clearly shows that a large region where the orbit attracts to an equilibrium point can be explained by this Lyapunov function (Fig.
		Note that the disappearance of this Lyapunov function is equivalent to the Hopf bifurcation; however, the occurrence of a Hopf bifurcation does not in general imply that a Lyapunov function existed prior to the bifurcation.
		Notice how the region where complex dynamics is forbidden by the existence of a Lyapunov function in (a) closely matches the region where the orbit attracts to an equilibrium point in the lower left of (b).
	sprott.physics.wisc.edu /chaos/lvmodel/joe/EC.doc (4034 words)

	syllabus (Site not responding. Last check: 2007-10-20)
		For the HQRB method, the main program is lyapunov.m, which is written as a MATLAB function file.
		The function lyapunov computes the trajectory and the Lyapunov characteristic exponents for discrete systems using the Householder QR Based Method.
		For the HQRBp1 and HQRBp2 methods the main functions are thqrbp1.m and thqrbp2 respectively.
	www.usc.edu /dept/engineering/mecheng/DynCon/readme.html (312 words)

	Chapter 5
		These two examples are discussed in this section and a rather technical definition of Lyapunov function (which applies to the damped pendulum and oscillator) is given.
		Exercises 1--3 involve checking that a given function is Lyapunov and using this function to sketch the phase plane.
		Exercises 4--11 study the damped pendulum using the energy as a Lyapunov function.
	math.bu.edu /odes/inst-manual/fed/ch5.html (1225 words)

	Brian Romanchuk, Paper Abstract (Site not responding. Last check: 2007-10-20)
		The classical means of achieving non-local results is the use of Lyapunov functions.
		Although powerful, the results are limited by the fact that one has to produce the correct Lyapunov function.
		In many cases, one almost has to have the problem solved in order to produce such a function; the function only serves to formalise the result.
	www-control.eng.cam.ac.uk /bgr/poly2.html (593 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Each element ; is the value of the scale-dependent Lyapunov ; exponent for a shell interval (see Shell_Bounds).
		Key outputs of this section: ; lyapunov, linear_limits (in the /Auto_Lin_Lim case).
		Default is NaN, ; which is the value this is set to if a Lyapunov expo- ; nent isn't well-defined.
	www.johnny-lin.com /idl_code/scale_lyapunov.pro (1636 words)

	PUBLICATIONS (Site not responding. Last check: 2007-10-20)
		Abstract- Critical condition is derived for a parameter-dependent quadratic Lyapunov function for a polytope of matrices.
		In this paper first design is proposed of a memory less feedback controller for uncertain system dynamic system with time varying delay.
		In this paper, we have studied a class of scaling functions, which satisfy criteria derived in the paper.
	www.geocities.com /sjain_d/PUBLICATION.html (215 words)

	Chapter 5
		Hamiltonian systems and systems with an integral are discussed in Section 5.3, and Section 5.4 considers systems with a Lyapunov function and gradient systems.
		Exercises 1-3 involve checking that a given function is Lyapunov and using this function to sketch the phase portrait.
		Exercises 4-11 study the damped pendulum using the energy as a Lyapunov function.
	math.bu.edu /odes/paul-inst-manual/sed/ch5.html (1405 words)

	LiapSolve (Site not responding. Last check: 2007-10-20)
		This algorithm casts the problem as a function minimization problem by use of a Lyapunov function for Nash equilibria.
		This is a continuously differentiable non negative function whose zeros coincide with the set of Nash equilibria of the game.
		If the objective function is larger than the tolerance, then the point is discarded.
	econweb.tamu.edu /gambit/doc/manual-0.97.0.6/x8559.html (331 words)

	On common quadratic Lyapunov functions for switched linear systems (Site not responding. Last check: 2007-10-20)
		Recent research on switched and hybrid systems has resulted in a renewed interest in determining conditions for the existence of a common quadratic Lyapunov function for a finite number of stable LTI systems.
		While efficient numerical solutions to this problem have existed for some time, compact analytical conditions for determining whether or not such a function exists for a finite number of systems have yet to be obtained.
		Our conditions also relate the existence of such a function to the stability boundary of the underlying switched linear system (thereby indicating that requiring the existence of such a function does not, in a certain sense, lead to overly conservative stability conditions).
	www-control.eng.cam.ac.uk /Seminars/abstracts/shorten.html (235 words)

	A Lyapunov Function for Tridiagonal Competitive-cooperative Systems
		A Lyapunov Function for Tridiagonal Competitive-cooperative Systems: SIAM Journal on Mathematical Analysis Vol.
		We construct a Lyapunov function for tridiagonal competitive-cooperative systems.
		The same function is a Lyapunov function for Kolmogorov tridiagonal systems, which are defined on a closed positive orthant in R
	epubs.siam.org /sam-bin/dbq/article/31614 (119 words)

	Weak Converse Lyapunov Theorems and Control-Lyapunov Functions (Site not responding. Last check: 2007-10-20)
		Weak Converse Lyapunov Theorems and Control-Lyapunov Functions: SIAM Journal on Control and Optimization Vol.
		Given a weakly uniformly globally asymptotically stable closed (not necessarily compact) set ${\cal A}$ for a differential inclusion that is defined on $\mathbb{R}^n$, is locally Lipschitz on $\mathbb{R}^n \backslash {\cal A}$, and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on $\mathbb{R}^n$.
		Using this result, we show that uniform global asymptotic controllability to a closed (not necessarily compact) set for a locally Lipschitz nonlinear control system implies the existence of a locally Lipschitz control-Lyapunov function, and from this control-Lyapunov function we construct a feedback that is robust to measurement noise.
	epubs.siam.org /sam-bin/dbq/article/39818 (156 words)

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